238 SIR WILLIAM THOMSON ON VORTEX MOTION. 



51. Both as regards the mathematical theory of the convergence of definite 

 integrals, and as illustrating the distribution of momentum in a fluid, it is inter- 

 esting to remark that, u denoting component velocity parallel to x, at any point 

 (x, y, z), the integral fffu dx dy dz, expressing momentum, may, as is readily 

 proved, have any value from — oo to +00 according to the portions of space 

 through which it is taken. 



52. As a last illustration of the distribution of momentum, let the containing 

 vessel be spherical of finite radius a. 



We have, as in § 19, 



9 = S + S x r + S 2 r 2 + &c, I 



\ ■ (14), 



+ T x r- 2 + T 2 r- 3 + &c.,j 



each series converging, provided r is less than a, and greater than the radius 

 of the smallest concentric spherical surface enclosing all the solids or vortices. 

 Now, by the condition that there be no flow across the fixed containing surface 

 we must have 



dtp 



which gives 



= 0, when r — a (15), 



S,- = — • — -MTl ( 16 ) 



a 



and (14) becomes 



^O + ^KK 1 ^)^- • • ^ 



But [§ 37 (1) ] if the whole amount of the ^-component of impulsive pressure 

 exerted by the fluid within the spherical surface of radius r, upon the fluid round 

 it be denoted by F, we have 



F = -ff<p cos 6d* . . . (18), 



6 being the inclination to OX of the radius through do-. Now cos 9 is a surface 

 harmonic of the first order, and therefore all the terms of the harmonic expan- 

 sion, except the first, disappear in the integral, which consequently becomes 



F =-( ,+2 :-')# T > cos «£ • ■ ■ (i9) - 



Now let 



Ti = _ Ax + By + Cz .... (20)> 



this being [Thomson & Tait, App. B, §§ i, j] the most general expression for a sur- 

 face harmonic of the first order. We have cos 6 = - ; and therefore (by spheri- 

 cal harmonics, or by the elementary analysis of moments of inertia of a uniform 

 spherical surface), 



